What is the relationship between Adjusted R-Square and R-Square?

Question

BlackTom AI · Accepted Answer

Question restatement: What is the relationship between Adjusted R-Square and R-Square?

Option A: Adjusted R-Square is some times less and some times more than R-Square
This claim suggests that the adjusted measure can swing above or below the unadjusted R-Squared depending on circumstances. In fact, the mathematical definition shows that adjusted R-Squared accounts for the number of predictors and the sample size, and it cannot exceed R-Squared because it penalizes the addition of variables that do not improve fit. Therefore, this option incorrectly implies that adjusted R-Squared can be greater than R-Squared in some cases.

Option B: Adjusted R-Square is always more than R-Square
This is contradicted by how adjusted R-Squared is constructed. The adjustment reduces the value when added predictors do not provide enough explanatory power relative to the penalty for extra parameters. This option overstates the relationship and is incorrect.

Option C: Adjusted R-Square is always equal to R-Square
Equality would occur only in a degenerate situation, such as when there are no predictors or when the model fits perfectly. In typical practice with one or more predictors, the adjusted measure differs from the unadjusted one. This option is not generally true.

Option D: Adjusted R-Square is always less than R-Square
This statement captures the standard relationship: adjusted R-Squared is R-Squared minus a penalty that increases with the number of predictors relative to the sample size. Because of the penalty term, adjusted R-Squared is typically lower than R-Squared, especially as model complexity grows. However, note that in a theoretical edge case (for example, when there are no predictors or perfect fit), they could be equal. The wording uses 'always,' which is a strict claim, but in practical and almost all typical scenarios, adjusted R-Squared is not greater than R-Squared.

Summary of reasoning across options:
- Option A is incorrect because adjusted R-Squared does not exceed R-Squared in general; it is not a mixed relationship in practice.
- Option B is incorrect because adjusted R-Squared is not typically larger than R-Squared; the penalty term prevents that in standard cases.
- Option C is incorrect since equality is not a general rule and only occurs in special degenerate situations.
- Option D is the closest to the standard interpretation, acknowledging that adjusted R-Squared is usually less than R-Squared due to the penalty for extra predictors, with the caveat that equality can occur in specific edge cases.

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